Estimation of the parameters of skew normal distribution by approximating the ratio of the normal density and distribution functions.

摘要

The normal distribution is symmetric and enjoys many important properties. That is why it is widely used in practice. Asymmetry in data is a situation where the normality assumption is not valid. Azzalini (1985) introduces the skew normal distribution reflecting varying degrees of skewness. The skew normal distribution is mathematically tractable and includes the normal distribution as a special case. It has three parameters: location, scale and shape. In this thesis we attempt to respond to the complexity and challenges in the maximum likelihood estimates of the three parameters of the skew normal distribution. The complexity is traced to the ratio of the normal density and distribution function in the likelihood equations in the presence of the skewness parameter. Solution to this problem is obtained by approximating this ratio by linear and non-linear functions. We observe that the linear approximation performs quite satisfactorily. In this thesis, we present a method of estimation of the parameters of the skew normal distribution based on this linear approximation. We define a performance measure to evaluate our approximation and estimation method based on it. We present the simulation studies to illustrate the methods and evaluate their performances.